Question

\[\int\frac{1}{x (3 + \log x)} dx\]

Answer 1

` Here, we  are" considering "log  x  as    log_e x . `
\[\text{Let I} = \int\frac{1}{x\left( 3 + \log x \right)}dx\]
\[\text{Putting }\log x = t\]
\[ \Rightarrow \frac{1}{x} = \frac{dt}{dx}\]
\[ \Rightarrow \frac{dx}{x} = dt\]
\[ \therefore I = \int\frac{dt}{3 + t}\]
\[ = \text{log }\left| 3 + t \right| + C\]
\[ = \text{log }\left| 3 + \text{log x }\right| + C \left[ \because t = \text{log x} \right]\]